6 The LDA+DMFT Approach Eva Pavarini Institute for Advanced Simulation Forschungszentrum J ¨ ulich GmbH Contents 1 The many-body problem 2 2 Low-energy models 7 3 Many-body models from DFT 10 3.1 Towards ab-initio Hamiltonians .......................... 10 3.2 Coulomb interaction tensor ............................ 12 3.3 Minimal material-specific models ........................ 14 4 Methods of solution 17 4.1 LDA+U ...................................... 17 4.2 LDA+DMFT ................................... 21 5 The origin of orbital order 26 6 Conclusions 30 A Constants and units 32 B Atomic orbitals 32 B.1 Radial functions .................................. 32 B.2 Real harmonics .................................. 32 B.3 Slater-Koster integrals .............................. 34 B.4 Gaunt coefficients and Coulomb integrals .................... 35 E. Pavarini, E. Koch, Dieter Vollhardt, and Alexander Lichtenstein The LDA+DMFT approach to strongly correlated materials Modeling and Simulation Vol. 1 Forschungszentrum J¨ ulich, 2011, ISBN 978-3-89336-734-4 http://www.cond-mat.de/events/correl11
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E. Pavarini, E. Koch, Dieter Vollhardt, and Alexander LichtensteinThe LDA+DMFT approach to strongly correlated materialsModeling and Simulation Vol. 1Forschungszentrum Julich, 2011, ISBN 978-3-89336-734-4http://www.cond-mat.de/events/correl11
It would indeed be remarkable if Nature fortified herself against further advances in knowledge
behind the analytical difficulties of the many-body problem. (Max Born, 1960)
1 The many-body problem
Most of chemistry and solid-state physics is described by the Hamiltonian
H = −1
2
∑
i
∇2i +
1
2
∑
i6=i′
1
|ri − ri′ |−
∑
i,α
Zα
|ri −Rα|−∑
α
1
2Mα∇2
α +1
2
∑
α6=α′
ZαZα′
|Rα −Rα′ | ,
where {ri} are the coordinates of the Ne electrons, {Rα} those of the Nn nuclei, Zα the atomic
numbers, andMα the nuclear masses. The Born-Oppenheimer product AnsatzΨ ({ri}, {Rα}) =ψ({ri}; {Rα})Φ({Rα}) simplifies the problem. The Schrodinger equation for the electrons,
Heψ = εψ, with
He = −1
2
∑
i
∇2i +
1
2
∑
i6=i′
1
|ri − ri′ |−∑
iα
Zα
|ri −Rα|+
1
2
∑
α6=α′
ZαZα′
|Rα −Rα′ |
= Te + Vee + Ven + Vnn, (1)
has however a simple solution only in the non-interacting limit (Vee = 0). In this case, He is
separable as He =∑
i h0e(ri) + Vnn, with
h0e(r) = −1
2∇2 −
∑
α
Zα
|r−Rα|= −1
2∇2 + vext(r).
In a crystal the external potential vext(r) is periodic, the eigenvectors of h0e(r) are Bloch func-
tions, ψnkσ(r). The eigenvalues are the corresponding band energies, εnk. Many-electron (Ne >
1) states may be obtained by filling energy levels εnk with electrons and anti-symmetrizing the
wave-function according to the Pauli principle (Slater determinant). For a half-filled band de-
scribed by the dispersion relation εk, such a Slater determinant has the form
ψ({ri}; {Rα}) =1√Ne!
ψk1↑(r1) ψk1↑(r2) . . . ψk1↑(rNe)
ψk1↓(r1) ψk1↓(r2) . . . ψk1↓(rNe)
......
......
ψkNe2
↑(r1) ψkNe2
↑(r2) . . . ψkNe2
↑(rNe)
ψkNe2
↓(r1) ψkNe2
↓(r2) . . . ψkNe2
↓(rNe)
. (2)
Unfortunately, the electron-electron repulsion is strong, and the non-interacting electrons ap-
proximation is insufficient to understand real materials. Because Vee is not separable, with
increasing Ne, finding the solution of the Schrodinger equation Heψ = εψ becomes quickly an
unfeasible task, even for a single atom.
A big step forward was the development of density-functional theory (DFT) [1, 2], described
in detail in the Lecture of Peter Blochl. DFT is based on the Hohenberg-Kohn theorem, which
LDA+DMFT 6.3
establishes the one-to-one correspondence between the ground-state electron density n(r) of an
interacting system and the external potential vext(r) acting on it. For any material described by
the Hamiltonian (1), the ground-state total energy is a functional of the electron density, E[n],
which is minimized by the ground-state density. E[n] can be written as
E[n] = F [n] +
∫
dr vext(r)n(r) + Enn = F [n] + V [n] + Enn.
F [n] = Te[n] + Eee[n], the sum of the kinetic and electron-electron interaction energy, is a
(unknown) universal functional (the same for all systems). V [n] is the system-specific potential
energy. The shiftEnn is the nucleus-nucleus interaction energy. The obstacle is that finding n(r)
still requires, in principle, the solution of the many-body problem (1). Kohn and Sham have
shown, however, that n(r) can be obtained by solving the Schrodinger equation of a fictitious
non-interacting system, whose external potential vR(r) is chosen such that the ground-state
density n0(r) equals n(r)
n(r) = n0(r) =occ∑
n
|ψn(r)|2.
To obtain the Hamiltonian h0e(r) of such an auxiliary problem we rewrite F [n] as
F [n] = T0[n] + EH [n] + Exc[n] = T0[n] +1
2
∫
dr
∫
dr′n(r)n(r′)
|r− r′| + Exc[n],
where T0[n] is the kinetic energy of the auxiliary system, EH [n] the classical electrostatic (or
Hartree) energy, and Exc[n] is the small exchange-correlation correction,
Exc[n] = Eee[n]− EH [n] + Te[n]− T0[n].
By minimizing the total energy with respect to {ψn}, with the constraint 〈ψn|ψn′〉 = δn,n′ , we
find the Kohn-Sham equation
h0e(r) ψn(r) = [−1
2∇2 + vR(r)]ψn(r) = εnψn(r). (3)
The eigenvalues εn are the Lagrange multipliers which enter the minimization through the con-
straint. The external (or reference) potential is given by
vR(r) = −∑
α
Zα
|r−Rα|+
∫
dr′n(r′)
|r− r′| +δExc[n]
δn.
The exchange-correlation functional is unknown, and includes a Coulomb (Eee[n] − EH [n])
and a kinetic energy (Te[n]− T0[n]) term. The latter can be transformed into a correction of the
Coulomb term by means of a coupling-constant integration: The interaction Vee is rescaled by
a parameter λ (with 0 ≤ λ ≤ 1), while keeping n(r) fixed; this constraint is fulfilled through
a reference potential vλR(r). Using the Hellmann-Feynman theorem to calculate ∂Eλ
∂λ, where
Eλ = 〈Ψλ|Hλ|Ψλ〉 is the ground-state energy at coupling constant λ, and then integrating over
λ to obtain E1 − E0, one may show that
Exc[n] =
∫
dr
∫
dr′n(r)n(r′)(g(r, r′)− 1)
|r− r′| ,
6.4 Eva Pavarini
where
g(r, r′) =
∫ 1
0
dλ gλ(r, r′).
The quantity n(r, r′) =∑
σ,σ′ n(rσ, r′σ′) = n(r′)n(r)gλ(r, r′) is the joint probability of finding
electrons at r and r′. The function gλ(r, r′) is the pair-correlation function. It can be shown that
gλ(r, r′)− 1 vanishes in the large |r− r′| limit.
In the Hartree-Fock approximation, in which the wavefunction is a Slater determinant, e.g. (2)
n(rσ, r′σ′) = n(rσ)n(r′σ′)− δσ,σ′
∣
∣
∣
∣
∣
∣
Ne/2∑
i
ψkiσ(r)ψkiσ(r′)
∣
∣
∣
∣
∣
∣
2
, (4)
where the last term accounts for the Pauli exclusion principle (exchange) and cancels the un-
physical interaction of each electron with itself (self-interaction) present in the Hartree energy.
The following sum rule holds for the pair-correlation function
∫
dr′ n(r′)(gλ(r, r′)− 1) = −1. (5)
This −1 is, in atomic units (Appendix A), a positive charge −e. The exchange-correlation en-
ergy Exc[n] may thus be interpreted as the energy gain due to the interaction of each electron
with an exchange-correlation hole with charge density n(r′)(gλ(r, r′)−1) surrounding it. Since
the exchange hole described by Eq. (4) already satisfy the sum rule (5), the remaining correla-
tion hole redistributes the charge density of the hole. In the one-electron case (Ne = 1), Exc[n]
merely cancels the Hartree self-interaction energy.
The main difficulty of DFT is to find good approximations to Exc[n]. The most common is
the local-density approximation (LDA), in which Exc[n] is replaced by its expression for a
homogeneous interacting electron gas with density equal to the local density n(r)
Exc[n] =
∫
drǫLDAxc (n(r))n(r). (6)
The LDA is particularly justified in systems with slowly varying spatial density n(r). For such
materials, we could split space into regions in which the density is basically constant and the
system can indeed be described by a homogeneous electron gas; if we add up the contributions
of all these regions of space we obtain the integral (6). The spin-polarized extension of the
local-density approximation is the local spin-density approximation (LSDA).
The ground-state electron-density n(r) can be obtained by solving (3) self-consistently. Vari-
ous successful methods have been developed to find the eigenvalues and eigenvectors of (3),
for solids and molecules. They are based on atomic-like orbitals (LMTO, NMTO), plane-
waves (pseudopotentials), combinations of both (LAPW, PAW), gaussians, or Green functions
(KKR) [3]. Through the years, DFT and the LDA have provided insight not only in solid-state
physics, but also in chemistry and even in systems of biological interest. For this reason DFT
became the standard model for electronic-structure calculations [1–3]. Strictly speaking, the
Kohn-Sham energies εn have no physical meaning except the highest occupied state, which
LDA+DMFT 6.5
0
0.5
1
-2 -1 0 1 2 3 4 5 6
4r2
|R
nl(r)
|2/a
B
r/aB
Cu F
0
0.5
1
-2 -1 0 1 2 3 4 5 6
4r2
|R
nl(r)
|2/a
B
r/aB
Cu F
0
0.5
1
-2 -1 0 1 2 3 4 5 6
4r2
|R
nl(r)
|2/a
B
r/aB
Cu F
0
0.5
1
-2 -1 0 1 2 3 4 5 6
4r2
|R
nl(r)
|2/a
B
r/aB
Cu F
Fig. 1: LDA solution of the Schrodinger equation for a single atom: 4πr2|Rnl(r)|2/aB as a
function of the distance from the nucleus, r/aB (atomic units, Appendix A). Blue: Cu 3d. Black:
Cu 4s. The 2p orbital of a F atom in r = 4aB is also shown (green). Cu has the electronic
configuration [Ar] 3d104s1 and F the configuration [He] 2s22p5.
yields the ionization energy, and their identification with one-particle energies is not justified.
The Kohn-Sham orbitals ψn(r) are just a tool to generate the ground-state density n(r). Never-
theless, in practice Kohn-Sham orbitals turned out to be very useful to explain the properties of
solids. Fermi surfaces, chemistry and many features of the electronic structure are qualitatively
and often quantitatively well described by DFT in the LDA approximation or its extensions.
The energy gap of semiconductors is underestimated, but can be corrected within many-body
perturbation theory (GW approximation, discussed in the Lecture of Karsten Held).
LDA fails to capture, however, the essential physics of strongly-correlated systems, even at a
qualitative level. At the center of this discrepancy are many-body effects between electrons in
open d or f shells. Since these electrons are very localized, the Coulomb repulsion between
them is significant. When Coulomb repulsion is strong, electrons lose their individuality: The
dynamics of a single electron depends on the position of all others, the Coulomb repulsion of
which it has to avoid (electrons are strongly correlated), and cannot be described by a refer-
ence mean-field potential. This happens for example in the case of Mott insulators. Because
the Kohn-Sham Hamiltonian (3) with the LDA exchange-correlation potential describes inde-
pendent electrons, many-electron states can be built from the Kohn-Sham orbitals as a single
Slater determinant. Thus, a non-magnetic crystal with an odd number of electrons per unit cell
has partially filled bands because of spin degeneracy, and therefore is metallic. However, due
to Coulomb repulsion, several transition-metal compounds with partially filled d shells are in-
sulating, paramagnetic above the Neel temperature TN , and sometimes exhibit a large gap. In
Fig. 1 the extensions of the atomic radial functions for the outer orbitals, 3d and 4s, of Cu can
be compared. While for 3d electrons the radial function decays very rapidly with distance, for
6.6 Eva Pavarini
x y
z
1 2
12
Rδ
δ=0.4%
γ=1
Iδ
δ=0-0.4%
δ=4.4%
c/2
l
R
γ=0.95
FCuK
1
1 2
(x,y,z) (y,x,-z)
γ=0.95
R
s
Fig. 2: Crystal structure, distortions and orbital order in KCuF3. Cu is at the center of F
octahedra enclosed in a K cage. The conventional cell is tetragonal with axes a, b, c. The
pseudocubic axes x, y, z pointing towards neighboring Cu are shown in the corner. Short (s)and long (l) CuF bonds alternate between x and y along all pseudocubic axes (co-operative
Jahn-Teller distortion). The distortions are measured by δ = (l− s)/(l+ s)/2 and γ = c/a√2.
R is the experimental structure (γ = 0.95, δ = 4.4%), Rδ (γ = 0.95) and Iδ (γ = 1) two
ideal structures with reduced distortions. In the I0 structure the cubic crystal-field at the Cu
site splits the 3d manifold into a t2g triplet and a eg doublet. In the R structure, site symmetry is
lowered further by the tetragonal compression (γ < 1) and the Jahn-Teller distortion (δ 6= 0).
The figure shows the highest energy d orbital. From Ref. [4].
4s electrons it is still sizable ∼ 2 A away from the nucleus, a typical interatomic distance in a
lattice. Thus in a crystal 4s electrons are likely to form delocalized states, while 3d electrons
tend to retain part of their atomic characteristics.
As example we take KCuF3. This system has a perovskite structure, shown in Fig. 2, with
each Cu surrounded by a F octahedron. The nominal valence for K, Cu and F is K+ (4s0), F−
(2p6), Cu2+ (3d94s0). The cubic crystal field at the Cu site splits the partially filled 3d levels
into the lower energy t2g (|xy〉, |xz〉, |yz〉), and the higher energy eg (|x2 − y2〉, |3z2 − r2〉)manifold; the electronic configuration is t62ge
3g. The co-operative Jahn-Teller distortion and the
tetragonal compression further reduce the site symmetry of Cu, and the eg doublet splits into
|3l2 − 1〉 and |s2 − z2〉. Because long (l) and short (s) CuF bond alternate between x and y
along all cubic axes, the highest energy d orbitals, |s2 − z2〉, form the pattern shown in Fig. 2.
The LDA band structure of KCuF3 is shown in Fig. 3. We can identify the bands from their
main character as F p-like (filled), Cu t2g-like (filled), Cu eg-like (occupied by 3 electrons),
Cu s- and K s-like (empty). The Fermi level is located in the middle of the eg-like bands.
LDA+DMFT 6.7
-8
-6
-4
-2
0
2
4
6
8
Z Γ X P N
ene
rgy
(eV
)
Fig. 3: LDA band structure of KCuF3. Blue: Cu eg-like bands. Red: Cu t2g-like bands. Black:
filled F p-like bands and empty bands.
Thus LDA predicts that KCuF3 is a metal, although it actually is an insulator (paramagnetic
down to TN = 40 K). A similar problem occurs in many other transition-metal compounds with
partially filled d shells: manganites, vanadates, titanates. This discrepancy cannot be solved
by simple improvements of the LDA functional. Coulomb repulsion effects beyond mean field
are essential to understand the origin of the insulating state in these materials. Other systems
for which similar considerations apply are heavy fermions and Kondo systems (f electrons) or
organics (molecular crystals).
2 Low-energy models
Lacking a working ab-initio theory, strongly-correlated systems have been studied for a long
time through low-energy model Hamiltonians. Within this approach only the states and inter-
actions believed to be most important to describe a given phenomenon are considered. Models
can be justified on the ground that at low energy, high-energy degrees of freedom can be, in
principle, projected out (downfolded), in the spirit of Wilson renormalization group. Their main
effect is assumed to be included implicitly in the low-energy model through a renormalization
of parameters. In LDA strongly-correlated transition-metal compounds usually have narrow d
bands close to the Fermi level (see Fig. 3) and thus the d bands, or a subgroup of those (eg-bands
for KCuF3) are believed to be the essential degrees of freedom. The minimal model to describe
a system with a narrow band at the Fermi level is the Hubbard model
H = −t∑
σ〈ii′〉c†iσci′σ + U
∑
i
ni↑ni↓ = H0 + U , (7)
6.8 Eva Pavarini
where c†iσ (ciσ) creates (destroys) an electron with spin σ at site i, niσ = c†iσciσ gives the i-
site occupancy per spin, t is the hopping integral between first neighbors, and U the on-site
Coulomb repulsion.
In the non-interacting limit (U = 0), the Hamiltonian (7) can be written in diagonal form
H0 =∑
kσ
εkc†kσckσ =
∑
kσ
εknkσ.
The band energy is given by εk = −t 1N
∑
〈ii′〉 eik·(Ri −Ri′), where Ri are lattice vectors, andN is
the number of sites; the operator c†kσ is the Fourier transform of c†iσ, i.e., c†
kσ = 1√N
∑
i eik·Ric†iσ,
and nkσ = c†kσckσ. At half-filling (Ne=N), the ground state is paramagnetic and metallic.
In the atomic limit (t = 0), the model (7) describes instead an insulating collection of indepen-
dent atoms with disordered magnetic moments.
Thus the Hubbard model captures the essence of the paramagnetic metal to paramagnetic insu-
lator (Mott) transition, and can qualitatively explain why systems like KCuF3 are paramagnetic
insulators in a large temperature range. Furthermore, it explains the fact that KCuF3 and most
strongly-correlated transition-metal compounds have an antiferromagnetic ground state. For
small t/U , by downfolding doubly occupied states, the Hubbard model (7) can be mapped onto
a spin 1/2-antiferromagnetic Heisenberg model
H → JAFM1
2
∑
〈ii′〉
[
Si · Si′ −1
4nini′
]
,
with coupling JAFM = 4t2/U . Thus at low temperature, when charge fluctuations play a minor
role, a transition to an antiferromagnetic state can take place. In strongly-correlated transition-
metal compounds, where the hopping t between correlated d states is mediated by the p orbitals
of the atom between two transition metals (e.g., F p states in KCuF3, Fig. 2), this many-body
exchange mechanism is called super-exchange. Because the Hubbard model can be solved
exactly only in special cases (e.g., in one dimension), it was for a long time impossible to
understand the nature of the Mott transition within this model. Understanding real materials
appeared even less likely. Progress came with the development of the dynamical mean-field
theory (DMFT) [5]. In DMFT, the Hubbard model, which describes a lattice of correlated sites,
is mapped onto an effective Anderson model, which describes a correlated impurity
Heff =∑
kσ
εknkσ + εd∑
σ
ndσ + Und↑nd↓ +∑
kσ
(Vkd c†kσdσ + V kd d
†σckσ).
Here d†σ (dσ) creates (destroys) an electron at the impurity site, and ndσ = d†σdσ counts the
number of electrons on the impurity; c†kσ (c
kσ) creates (destroys) a bath electron with energy
εk, and Vkd is the hybridization between bath and impurity. This auxiliary quantum-impurity
model is solved self-consistently. The solution is found when the interacting Green function
G(ω) of the auxiliary model equals the local Green function Gii(ω) of the Hubbard model (7)
G(ω) = Gii(ω) =1
Nk
∑
k∈BZ
1
ω + µ− εk −Σ(ω)=
∫
dερ(ε)
ω + µ− ε−Σ(ω). (8)
LDA+DMFT 6.9
Here µ is the chemical potential, the sum is over Nk k-points of the Brillouin Zone (BZ), Σ(ω)
is the self-energy of the quantum impurity model and ρ(ε) is the density of states. The self-
energy Σ(ω) can be obtained from the Dyson equation of the impurity problem
G−1(ω) = G−1(ω) +Σ(ω), (9)
where G(ω) is the non-interacting Green function of the Anderson model (bath Green function).
The main approximation in DMFT consists in neglecting spatial fluctuations in the lattice self-
energy; this approximation becomes exact in the limit of infinite coordination number [5]. The
Anderson model is a full many-body Hamiltonian, known since long in the framework of the
Kondo effect [6], but, in contrast to the original Hubbard model, it describes only a single cor-
related site. It can be solved numerically with different approaches (quantum-impurity solvers):
the numerical renormalization group [6], various flavors of quantum Monte Carlo (QMC) [7,8],
Lanczos [9], or other methods [6, 10]. Some of the most important solvers are presented in the
Lectures of Erik Koch, Nils Blumer, and Philipp Werner. If we use QMC, we have to work
in imaginary time/frequencies, and replace the frequency ω in (8, 9) with iωn, where ωn are
Fermionic Matsubara frequencies, ωn = (2n+ 1)πkBT , and T is the temperature.
The DMFT approach is discussed in detail in the Lecture of Marcus Kollar. We recall here
some important conclusions obtained by studying the half-filled Bethe lattice, described for
U = 0 by a semi-elliptical density of states [10]. In the Fermi-liquid regime (metallic phase,
low temperature, ω ∼ µ = 0), the self-energy can be expanded as
Σ(ω + i0+) ∼ U
2+ (1− 1/Z)ω − i∆ω2 + . . . .
The effective mass of quasi-particles is m∗ = m/Z and their life-time ∝ 1/∆; Z is the quasi-
particle weight. In the Mott insulating regime, the self-energy has instead the following low-
frequency behavior
Σ(ω + i0+) ∼ U
2+ Γ/ω − iπΓδ(ω) + . . . ,
where Γ can be viewed as an order parameter. Thus the real part of the self-energy diverges
at ω ∼ 0; the strong ω dependence of Σ(ω) is essential to obtain the Mott metal-insulator
transition in the one-band Hubbard model.
The model Hamiltonian approach has proven effective in gaining insight into the behavior of
strongly-correlated systems. However, the actual derivation of low-energy models by down-
folding the full many-body problem, although formally possible, is in practice unfeasible and
would in general lead to complex interactions beyond the Hubbard model [11]. The insight
is thus gained at the price of neglecting all interactions that are thought not to have a direct
influence on the specific phenomenon, and then relating the few free parameters (here t and U)
to experimental data. It is clear that simple models such as the Hubbard model (7), although
grasping an essential aspect of Mott physics, are hardly sufficient to describe the complexity of
real materials such as KCuF3. Thus they have been extended to include many orbitals (e.g., the
full d shell), crystal-field splittings (which divides the d shell, e.g., into 3-fold degenerate t2g
6.10 Eva Pavarini
and 2-fold degenerate eg states), multiplets (whose description requires taking into account the
full Coulomb interaction tensor and the spin-orbit interaction), lattice distortions (which, in the
case of KCuF3, split the eg and the t2g manifold and change the hopping integrals), filled (e.g.,
F p in KCuF3) or excited (Cu s, K s, . . . ) states, and Coulomb repulsion between neighbors.
As we will see later in some examples, these details do matter when we want to understand
real materials; neglecting them easily leads to wrong conclusions. Some of the parameters of
such extended Hubbard models can indeed be obtained by fitting to experiments, but with the
increasing number of free parameters it becomes impossible to put any theory to a real test.
3 Many-body models from DFT
3.1 Towards ab-initio Hamiltonians
The dream of calculating the parameters of model Hamiltonians ab-initio exists since long.
We know from the successes of LDA that the Kohn-Sham orbitals obtained within the local-
density approximation carry the essential information about the structure and chemistry of a
given material. It appears therefore natural to build material-specific many-body models starting
from the LDA. This can be achieved by constructing ab-initio a basis of localized LDA Wannier
functions
ψinσ(r) =1√N
∑
k
e−iRi·k ψnkσ(r),
and the corresponding many-body Hamiltonian, which is the sum of an LDA term HLDA, a
Coulomb term U , and a double-counting correction HDC
He = HLDA + U − HDC. (10)
The LDA part of the Hamiltonian is given by
HLDA = −∑
σ
∑
in,i′n′
ti,i′
n,n′c†inσci′n′σ, (11)
where c†inσ (cinσ) creates (destroys) an electron with spin σ in orbital n at site i, and
ti,i′
n,n′ = −∫
drψinσ(r)[−1
2∇2 + vR(r)]ψi′n′σ(r). (12)
The on-site (i = i′) terms yield the crystal-field matrix while the i 6= i′ contributions are the
hopping integrals. The Coulomb interaction U is given by
U =1
2
∑
ii′jj′
∑
σσ′
∑
nn′pp′
U iji′j′
np n′p′c†inσc
†jpσ′cj′p′σ′ci′n′σ,
with
U iji′j′
np n′p′ = 〈inσ jpσ′|U |i′n′σ j′p′σ′〉 (13)
=
∫
dr1
∫
dr2 ψinσ(r1)ψjpσ′(r2)1
|r1 − r2|ψj′p′σ′(r2)ψi′n′σ(r1).
LDA+DMFT 6.11
To avoid double counting, the correction HDC should cancel the electron-electron interaction
contained in HLDA, which in (10) is explicitly described by U . However, it is more meaningful
to take advantage of the LDA, and subtract from U the long-range Hartree and the mean-field
exchange-correlation interaction, described by the LDA; from the successes of the LDA we
expect that they are well accounted for. Thus the difference U − HDC is a short-range many-
body correction to the LDA.
The Hamiltonian (10) still describes the complete many-body problem, finding the solution of
which remains an impossible task. To make progress, we separate the electrons in two types,
the correlated or heavy electrons and the uncorrelated or light electrons. For the correlated
electrons LDA fails qualitatively and U − HDC has to be taken into account explicitly; we
can assume that U − HDC is local (on-site) or almost local (between near neighbors). For the
uncorrelated electrons the LDA is overall a good approximation, and we do not consider any
correction U − HDC. By truncating U − HDC to the correlated sector we implicitly assume
that the effect of light electrons is a mere renormalization or screening of Coulomb parameters
in the correlated sector. This implies that the Coulomb couplings cannot be calculated directly
from (13). The calculation of screened Coulomb integrals remains to date a major challenge,
and effects beyond Coulomb parameter renormalization are usually neglected. Since correlated
electrons partially retain their atomic character, in the rest of this Lecture we will label them
with the quantum numbers lmσ, as in the atomic limit. To first approximation we can assume
that U − HDC is local and that correlated electrons belong to a single shell (e.g., d electrons,
l = 2). Thus we can write the Hamiltonian as the generalized Hubbard model
He = HLDA + U l − H lDC, (14)
where the screened Coulomb interaction is
U l =1
2
∑
i
∑
σσ′
∑
mαm′α
∑
mβm′β
Umαmβm′αm
′βc†imασc
†imβσ′cim′
βσ′cim′
ασ, (15)
and H lDC is, in principle, the mean-field value of U l. More generally, we can include in the
Hamiltonian (14) the Coulomb interaction between first neighbors, different shells, etc. Al-
though (14) is simpler than the original model (10), it still describes a full many-body problem;
however, as we will see later, such a problem can be solved numerically within the dynamical
mean-field approximation.
At the heart of (14) is the assumption that the U l − H lDC is local, or almost local; thus it
is essential to use a localized basis in the correlated sector, a basis in which the separation of
heavy and light electrons makes actually sense. Localized Wannier functions can be constructed
in different ways. Successful methods are the ab-initio downfolding procedure based on the
NMTO approach [12] and the maximally-localized Wannier functions algorithm of Marzari
and Vanderbilt [13]; a lighter alternative to localized Wannier functions are projected local
orbitals [14]. The latter two methods are presented in the Lecture of Jan Kunes.
6.12 Eva Pavarini
3.2 Coulomb interaction tensor
The screened Coulomb interaction, central to building the Hamiltonian (14), has the same form
as the bare Coulomb interaction tensor. To identify the different terms in U l, it is useful to
derive the bare Coulomb integrals for atomic orbitals ψnlm(r) = Rnl(r)Ylm(θ, φ) (see Appendix
B). The generalization to Wannier orbitals is straightforward. First, we write the electronic
positions in spherical coordinates, ri = ri(sin θi cosφi, sin θi sin φi, cos θi), and express the
Coulomb interaction as
1
|r1 − r2|=
∞∑
k=0
rk<rk+1>
4π
2k + 1
k∑
q=−k
Y kq (θ2, φ2)Y
k
q (θ1, φ1), (16)
where r< ( r>) is the smaller (larger) of r1 and r2. By inserting (16) into (13) we obtain
Umαmβm′αm
′β=
2l∑
k=0
ak(mαm′α, mβm
′β)Fk,
where ak are angular integrals
ak(mαm′α, mβm
′β) =
4π
2k + 1
k∑
q=−k
〈lmα|Y kq |lm′
α〉〈lmβ|Yk
q |lm′β〉,
and Fk radial Slater integrals
Fk =
∫
dr1 r21
∫
dr2 r22 R
2nl(r1)
rk<rk+1>
R2nl(r2).
The most important Coulomb integrals are the two-index terms: the direct (Umm′mm′) and ex-
change (Umm′m′m, with m 6= m′) integrals, which can be expressed as
Umm′mm′ = Um,m′ =
2l∑
k=0
ak(mm,m′m′)Fk,
Umm′m′m = Jm,m′ =2l∑
k=0
ak(mm′, m′m)Fk.
It can be shown that Um,m′ and Jm,m′ are positive, and that Um,m′ ≥ Jm,m′ . If we neglect
all terms but the direct and the exchange Coulomb interaction, only density-density terms (∝nimσnim′σ′ , with nimσ = c†imσcimσ) remain, and the Coulomb interaction takes a simpler form
U l ∼ 1
2
∑
iσ
∑
mm′
Um,m′ nimσnim′-σ +1
2
∑
iσ
∑
m6=m′
(Um,m′ − Jm,m′)nimσnim′σ. (17)
The contributions neglected in (17), spin-flip exchange terms (∝ Jm,m′) and off-diagonal con-
tributions (terms with more than two different orbital indices), are important to get the correct
multiplet structure. They are often neglected in DMFT calculations based on QMC solvers
because they can generate a strong sign problem.
LDA+DMFT 6.13
In electronic-structure calculations real harmonics (Appendix B) rather than spherical harmon-
ics are normally used; therefore here we will give the Coulomb integrals for d electrons in the
basis of real harmonics. It is useful to introduce first average Coulomb parameters
Uavg =1
(2l + 1)2
∑
m,m′
Um,m′ = F0,
Uavg − Javg =1
2l(2l + 1)
∑
m,m′
(Um,m′ − Jm,m′).
For d-electrons, only F0, F2 and F4 contribute to the Coulomb integrals, and we can show that
Javg = (F2 + F4)/14. For hydrogen-like 3d orbitals, F4/F2 = 15/23, while for realistic 3d
orbitals this ratio is slightly smaller. A typical value is F4/F2 ∼ 0.625 = 5/8.
Fig. 10: Evolution of crystal structure and LDA band structure in the series of 3d1 perovskites
with the GdFeO3-type distortion. From Ref. [12].
The LDA+DMFT scheme can be easily extended to treat clusters, by using a supercell and
treating the supercell as impurity; alternative extensions to account for the k dependence of the
self-energy are the dynamical-cluster approximation (DCA) [23], the dual-fermion approach,
or GW+DMFT. Some of these methods will be discussed in the Lectures of Sasha Lichtenstein
and Karsten Held. Finally, LDA+DMFT, as LDA+U, can be also made charge self-consistent.
This requires to work, as in LDA+U, with the full Hamiltonian and to account explicitly for the
double-counting correction.
The calculation of the screened Coulomb parameters is a major open problem, in LDA+DMFT
as in LDA+U. In the absence of a definitive method, a useful approach is to analyze trends in
similar materials, to single out the effects of chemistry and structural distortions from those
of the Coulomb interaction. We adopt this approach in Ref. [24] to study the Mott transition
in the series of 3d1 (t12ge0g) perovskites. These materials all have the GdFeO3-type structure,
with distortions (tilting, rotation, deformation of the cation cage) that increase along the series
(Fig. 10). By means of massive downfolding based on the NMTO method [12], we obtain
the material-specific t2g Wannier basis (crystal-field orbitals) shown in Fig. 5. With increasing
distortions, the t2g band-width decreases and the crystal-field splitting increases, reaching ∼300 meV in YTiO3, still a small fraction of the t2g band width. We find that, despite of its small
value, the crystal-field splitting plays a crucial role in helping the metal-insulator transition, by
reducing the orbital degeneracy [25] of the many-body states and favoring the formation of an
orbitally-ordered state.
6.26 Eva Pavarini
Fig. 11: Orbital-order (empty orbital) in the 3d2 perovskites LaVO3 and YVO3. From Ref. [26].
5 The origin of orbital order
In this Section we will show an example of how LDA+DMFT can be used as a tool to understand
physical phenomena. Orbital order is believed to play a crucial role in determining the electronic
and magnetic properties of many transition-metal compounds. Still, the origin of orbital order
in real materials is a subject of hot debate.
The hallmark of orbital order is the co-operative Jahn-Teller distortion. A paradigmatic example
is KCuF3. The co-operative Jahn-Teller distortion is shown in Fig. 2. This static distortion
gives rise to a crystal field, which splits the otherwise degenerate eg doublet. LDA+DMFT
calculations have proven that, due to Coulomb repulsion, even a crystal-field splitting much
smaller than the band width can lead to orbital order. The importance of such effect for real
materials has been realized first for LaTiO3 and YTiO3 [24, 12]. The same effect is at work
in a number of other systems with different electronic structure. We discuss few cases. In
3d2 vanadates [26] the t2g crystal-field splitting is even smaller than in 3d1 perovskites. Still,
orbital fluctuations are already strongly suppressed at room temperature, yielding the orbital-
order shown in Fig. 11. In Ca2RuO4, due to the layered perovskite structure, the 2/3-filled t2gbands split into a wide xy and two narrow xz and yz bands. In the low-temperature phase (S-
Pbca) the system is an insulator with a small gap and exhibits xy-orbital order: at each site the
xy orbital is filled with two electrons. Above 350 K, in the L-Pbca phase, Ca2RuO4 is metallic
and no orbital order has been reported. We find (Fig. 9) that the metal-insulator transition is
driven by the structural L-Pbca → S-Pbca phase transition; furthermore, in the insulating phase
the ∼ 300 meV crystal-field splitting overcomes the band missmatch and we find xy orbital
order [22]. The case of 3d9 KCuF3 and 3d4 LaMnO3 is even more extreme: the eg crystal-field
splitting is ∼ 0.5−1 eV at 300 K; with such a large splitting, orbital fluctuations are suppressed
up to melting temperature.
LDA+DMFT 6.27
0
1
0 500 1000 1500
p (D
MF
T)
T / K
R
R0.4
I0.4
I0.2I0.1I0.05I0 I0
0
1
0 500 1000 1500
p (D
MF
T)
T / K
R
R0.4
I0.4
I0.2I0.1I0.05I0 I0
0
1
0 500 1000 1500
p (D
MF
T)
T / K
R
R0.4
I0.4
I0.2I0.1I0.05I0 I0
Fig. 12: Orbital order transition in KCuF3. Orbital polarization p as a function of temperature
calculated in LDA+DMFT. R: U = 7 eV, experimental structure. Circles: U = 7 eV, idealized
structures Rδ and Iδ with decreasing crystal-field. Triangles: U = 9 eV, I0 only. Squares:
two-sites CDMFT. From Ref. [4].
Orbital order can already arise, however, in the absence of static distortions. In a seminal
work, Kugel and Khomskii [27] showed that in strongly-correlated systems with orbital degrees
of freedom (degenerate eg or t2g levels), many-body effects can give rise to orbital order via a
purely electronic mechanism (spin and orbital super-exchange). In this picture, the co-operative
Jahn-Teller distortion (and thus the crystal-field splitting) is a consequence of orbital order. In
the opposite scenario, the co-operative Jahn-Teller distortion is due to the electron-phonon cou-
pling, which removes orbital degeneracy, and orbital order is driven by the static distortion,
as discussed above. We analyze these two scenarios for KCuF3 and LaMnO3, the text-book
examples [28] of orbitally-ordered systems. LDA+U total energy calculations show [15, 29]
that in these systems the co-operative Jahn-Teller distortion is stabilized by U , a result recently
confirmed in LDA+DMFT [30]. This could indicate that super-exchange is the driving mecha-
nism. However, if this is the case it is hard to explain why the magnetic transition temperature,
determined by super-exchange, is much lower than the orbital-order transition temperature:
TN ∼ 40 K for KCuF3 and TN ∼ 140 K for LaMnO3, while the co-operative Jahn-Teller
distortion persists up to 1000 K or more.
LaMnO3 and KCuF3 can both be described by a two-band eg Hubbard model. In the case
of LaMnO3 we have additionally to take into account the Hund’s rule coupling between egelectrons and t2g spins, St2g . Thus the minimal model to understand orbital order in these two
6.28 Eva Pavarini
x y
z
Fig. 13: Orbital order (LDA+DMFT calculations) in the rare-earth perovskite TbMnO3 with
the GdFeO3-type structure. From Ref. [32]. This system has the same structure of LaMnO3.
systems is the Hamiltonian [31]
H = −∑
imσ,i′m′σ′
ti,i′
m,m′ui,i′
σ,σ′c†imσci′m′σ′ − h
∑
im
(nim↑ − nim↓)
+ U∑
im
nim↑nim↓ +1
2
∑
im( 6=m′)σσ′
(U − 2J − Jδσ,σ′)nimσnim′σ′ .
Here m,m′ = 3z2 − r2, x2 − y2. The local magnetic field h = JSt2g describes the Hund’s rule
coupling to t2g electrons, and uiσ,i′σ′ = 2/3(1 − δi,i′) accounts for the disorder in orientation
of the t2g spins. In the case of KCuF3 uiσ,i′σ′ = δσ,σ′ and h = 0. For the Coulomb parameters
we use the theoretical estimates J = 0.9 eV (KCuF3) and J = 0.75 eV (LaMnO3) and vary
U around 5 eV (LaMnO3) and 7 eV (KCuF3). In the high-spin regime, our results are not very
sensitive to h; we show results for h ∼ 1.3 eV. We use the massive downfolding technique
based on the NMTO method to calculate hopping integrals and crystal-field splittings.
To single out the effects of many-body super-exchange (Kugel-Khomksii mechanism) from the
effect of the crystal-field splitting, we perform LDA+DMFT and LDA+CDMFT calculations for
a series of hypothetical structures, in which the distortions (and thus the crystal-field splitting)
are progressively reduced. In the case of KCuF3, these hypothetical structures are shown in
Fig. 2, and the corresponding eg bands are shown in Fig. 8. For each structure we calculate
the order parameter, the orbital polarization p, defined as the difference in the occupations of
natural orbitals. In Fig. 12 we show p as a function of temperature. For the experimental
structure, p(T ) ∼ 1 till melting temperature; this means that, if the structure stays the same,
the system remains orbitally ordered till the crystal melts. The empty orbitals on different sites
make the pattern shown in Fig. 2. For the ideal cubic structure I0, we find that p(T ) = 0 at
high temperature, but a transition occurs at TKK ∼ 350 K. This TKK is the critical temperature
LDA+DMFT 6.29
0 500 1000
90o
150o
60o
180o
θ (
DM
FT
)
T/K
R0
I0 R2.4800K
R6
R11
0 500 1000
90o
150o
60o
180o
θ (
DM
FT
)
T/K
R0
I0 R2.4800K
R6
R11
0 500 1000
90o
150o
60o
180o
θ (
DM
FT
)
T/K
R0
I0 R2.4800K
R6
R11
0 500 1000
90o
150o
60o
180o
θ (
DM
FT
)
T/K
R0
I0 R2.4800K
R6
R11
0 500 1000
90o
150o
60o
180o
θ (
DM
FT
)
T/K
R0
I0 R2.4800K
R6
R11
0 500 1000
90o
150o
60o
180o
θ (
DM
FT
)
T/K
R0
I0 R2.4800K
R6
R11
0 500 1000
90o
150o
60o
180o
θ (
DM
FT
)
T/K
R0
I0 R2.4800K
R6
R11
0 500 1000
90o
150o
60o
180o
θ (
DM
FT
)
T/K
R0
I0 R2.4800K
R6
R11
0 500 1000
90o
150o
60o
180o
θ (
DM
FT
)
T/K
R0
I0 R2.4800K
R6
R11
0 500 1000
90o
150o
60o
180o
θ (
DM
FT
)
T/K
R0
I0 R2.4800K
R6
R11
0
1
0 500 1000
p (
DM
FT
)
T/K
I0I0
I0I0 R0
R2.4800K
R6R11
0
1
0 500 1000
p (
DM
FT
)
T/K
I0I0
I0I0 R0
R2.4800K
R6R11
0
1
0 500 1000
p (
DM
FT
)
T/K
I0I0
I0I0 R0
R2.4800K
R6R11
0
1
0 500 1000
p (
DM
FT
)
T/K
I0I0
I0I0 R0
R2.4800K
R6R11
0
1
0 500 1000
p (
DM
FT
)
T/K
I0I0
I0I0 R0
R2.4800K
R6R11
0
1
0 500 1000
p (
DM
FT
)
T/K
I0I0
I0I0 R0
R2.4800K
R6R11
0
1
0 500 1000
p (
DM
FT
)
T/K
I0I0
I0I0 R0
R2.4800K
R6R11
0
1
0 500 1000
p (
DM
FT
)
T/K
I0I0
I0I0 R0
R2.4800K
R6R11
0
1
0 500 1000
p (
DM
FT
)
T/K
I0I0
I0I0 R0
R2.4800K
R6R11
0
1
0 500 1000
p (
DM
FT
)
T/K
I0I0
I0I0 R0
R2.4800K
R6R11
Fig. 14: Orbital order transition in LaMnO3. Orbital polarization p (left) and (right) occupied
state |θ〉 = cos θ2|3z2 − r2〉 + sin θ
2|x2 − y2〉 as a function of temperature. Solid line: 300 K
experimental structure (R11) and 800 K experimental structure. Dots: orthorhombic structures
with half (R6) or no (R0) Jahn-Teller distortion. Pentagons: 2 (full) and 4 (empty) sites CDMFT.
Dashes: ideal cubic structure (I0). Circles: U = 5 eV. Diamonds: U = 5.5 eV. Triangles:
U = 6 eV. Squares: U = 7 eV. Crystal field splittings (meV): 840 (R11), 495 (R6), 168 (R800 K2.4 ),
and 0 (I0). From Ref. [33].
in the absence of Jahn-Teller effect. Our result shows that around 350 K super-exchange could
drive alone the co-operative Jahn-Teller distortion. However, experimentally, the co-operative
Jahn-Teller distortion persists up to 800 K or even higher temperature. TKK, although large, is
not large enough to explain the presence of a co-operative Jahn-Teller distortion above 350 K;
electron-phonon interaction plays a key role. Fig. 12 shows that a ∼ 200 meV crystal-field (as
in the ideal R0.4, which has a Jahn-Teller distortion ∼ 10% of the experimental structure) yields
already an almost complete suppression of orbital fluctuations up to at least 1500 K.
In the case of LaMnO3 we find (see Fig. 14) TKK ∼ 700 K. However, besides the co-operative
Jahn-Teller distortion and tetragonal compression, LaMnO3 exhibits a GdFeO3-type distortion
(Fig. 13), which tends to reduce the eg band-width [12]. Thus we study in addition an ideal
structure R0 with all distortions except the Jahn-Teller. For such system we cannot obtain TKK
from p(T ), because, due to the crystal-field splitting ∼ 200 meV, Coulomb repulsion strongly
suppress orbital fluctuations even at 1500 K. Instead, we study the evolution of the occupied
orbital |θ〉 = cos θ2|3z2 − r2〉+ sin θ
2|x2 − y2〉 with temperature. For the experimental structure
(R11) we find θ ∼ 108o, in agreement with experiments, and for the I0 structure we obtain
θ = 90o. For the R0 structure we find two regimes. At high temperature the occupied orbital
is the lower energy crystal-field orbital (θ = 180o). At TKK ∼ 550 K super-exchange rotates
this θ towards 90o, reaching 1300 in the zero temperature limit. Such TKK is still very large,
but again not sufficient to explain that the Jahn-Teller distortion persist in nanoclusters up to
6.30 Eva Pavarini
melting temperature [34]. Thus, as for KCuF3, electron-lattice coupling is essential to explain
the co-operative Jahn-Teller distortion at high temperature.
Although the co-operative Jahn-Teller distortion persists in domains up to melting temperature,
a order-to-disorder (or orbital melting) transition has been reported at TOO ∼ 750 K [35]. Since
TKK ∼ TOO, super-exchange could play a crucial role in such transition. To resolve this is-
sue, we analyze a series of materials for which TOO has been measured: rare-earth manganites
with structures similar to LaMnO3. For this series, it has been reported that TOO strongly in-
creases with decreasing the rare-earth radius, reaching about 1500 K in TbMnO3. Instead, with
LDA+DMFT and actually find the opposite trend: TKK is maximum in LaMnO3 and slightly
decreases along the series. Taking the tetragonal crystal-field into account reduces TKK, further
increasing the discrepancies with experiment. This proves that, surprisingly, super-exchange
effects, although very efficient, in the light of the experimentally reported trends, play a minor
role for the orbital order melting observed in rare earth manganites [32].
6 Conclusions
The many-body problem is central to theoretical solid-state physics. Density functional the-
ory is the standard approach for describing the electronic properties of materials. It is a very
successful method, which allows us to understand and predict the properties of many systems.
However, DFT practice fails completely for strongly-correlated materials, in which the move-
ment of one electron depends on the actual, not only the mean position of all other electrons,
since it has to avoid their Coulomb repulsion. In this Lecture we have seen a successful scheme
to deal with strongly-correlated systems: LDA+DMFT. It is based on the separation of electrons
into correlated and uncorrelated. While for uncorrelated electrons we use standard methods
based on density-functional theory, for correlated electrons we build material-specific many-
body models and solve them with DMFT. Building models requires the construction of local-
ized and material-specific basis sets. To this end, various successful approaches have been
devised, such as the ab-initio downfolding technique, maximally-localized Wannier functions
and projectors. Material-specific many-body models are however complex. Solving them
with DMFT requires flexible and efficient quantum-impurity solvers. Examples are Hirsch-
Fye QMC, continuous-time QMC, and Lanczos. Short and/or long range spatial correlations
can be in principle accounted for within different methods: CDMFT, DCA, dual fermions,
GW+DMFT. Thanks to the improvements of impurity solvers and to modern supercomputers,
we have been able to include in LDA+DMFT calculations more degrees of freedom, to reach
experimental temperatures, to calculate properties beyond the spectral function, and to move
towards predictive power. From the many successful applications of LDA+DMFT we have
learned that details do matter; for example a crystal-field splitting of merely hundred meV, typi-
cally neglected in studies based on simple model Hamiltonians, plays a crucial role in stabilizing
the orbitally-ordered Mott insulating state [24], and super-exchange, although very efficient, is
not the driving mechanism of orbital order in the text-book examples of actual orbitally-ordered
materials, LaMnO3 and KCuF3.
LDA+DMFT 6.31
Much work lies ahead. A general and effective quantum-impurity solver is not yet available;
effective k-dependent extensions of LDA+DMFT are under development. Central to the success
of LDA+DMFT and its extensions is the separation of electrons into light and heavy. This
separation, however, is also the main source of trouble. The double-counting correction is
essentially unknown. To account for screening effects in a realistic setting is very difficult.
Furthermore, by truncating the Coulomb interaction outside the correlated sector, we assume
implicitly that the effect of many-body downfolding to the correlated sector is only screening;
effects which go beyond the mere Coulomb renormalization are usually neglected. Ultimately,
these approximations have to be put to a test.
Finally, we have to remember that many-body phenomena are emergent behaviors [36]. Each of
such phenomena, although in principle described by the same many-body Hamiltonian (1), the
theory of almost everything [37], may have a very different nature. To predict new phenomena
before they are observed is therefore extremely challenging. After they are discovered, they are
often elusive and might remain mysterious for decades, like it is happening for high-temperature
superconductivity. In developing ab-initio theories for strongly-correlated systems, we have
thus always to keep in mind that the crucial aspect to explain a given phenomenon might be
hidden in some detail which in more ordinary circumstances would play no role. The challenge
is thus to identify which details do matter.
Acknowledgment
Support of the Deutsche Forschungsgemeinschaft through FOR1346 is gratefully acknowledged.
6.32 Eva Pavarini
Appendices
A Constants and units
In this Lecture, formulas are written in atomic units. The unit of mass m0 is the electron mass
(m0 = me), the unit of charge e0 is the electron charge (e0 = e), the unit of length a0 is the
Bohr radius (a0 = aB ∼ 0.52918 = A), and the unit of time is t0 = 4πε0~a0/e2. In these units,
me, aB , e and 1/4πε0 have the numerical value 1; the speed of light is c = 1/α ∼ 137 in atomic
units. The unit of energy is 1Ha = e2/4πε0a0 ∼ 27.211 eV. These are the natural units for
theory. When comparing to experiments, for convenience, we give the energies in eV or meV.
B Atomic orbitals
B.1 Radial functions
The nlm hydrogen-like atomic orbital is given by
ψnlm(ρ, θ, φ) = Rnl(ρ)Yml (θ, φ),
whereRnl(ρ) is the radial function and Y lm(θ, φ) a spherical harmonic, ρ = Zr and Z the atomic
number. In atomic units, the radial functions are
Rnl(ρ) =
√
(
2Z
n
)3(n− l − 1)!
2n[(n + l)!]3e−ρ/n
(
2ρ
n
)l
L2l+1n−l−1
(
2ρ
n
)
,
where L2l+1n−l−1 are generalized Laguerre polynomials of degree n− l − 1.
The radial function for n = 1, 2, 3 are
R1s(ρ) = 2 Z3/2 e−ρ
R2s(ρ) =1
2√2Z3/2 (2− ρ) e−ρ/2
R2p(ρ) =1
2√6Z3/2 ρ e−ρ/2
R3s(ρ) =2
3√3Z3/2 (1− 2ρ/3 + 2ρ2/27) e−ρ/3
R3p(ρ) =4√2
9√3Z3/2 ρ(1− ρ/6) e−ρ/3
R3d(ρ) =2√2
81√15Z3/2 ρ2 e−ρ/3
where we used the standard notation s for l = 0, p for l = 1 and d for l = 2.
B.2 Real harmonics
To study solids, it is usually convenient to work in the basis of real harmonics. The latter are
defined in terms of the spherical harmonics as follows:
yl0 = Y l0 , ylm =
1√2(Y l
−m + (−1)mY lm), yl−m =
i√2(Y l
−m − (−1)mY lm), m > 0.
LDA+DMFT 6.33
y
x
z
Fig. 15: The s (first row), py, pz, px (second row), and dxy, dyz, d3z2−r2 , dxz, dx2−y2 (last row)
real harmonics.
Using the definitions x = r sin θ cosφ, y = r sin θ sin φ, z = r cos θ, we can express the
l = 0, 1, 2 real harmonics (Fig. 15) as
s = y00 = Y 00 =
√
14π
py = y1−1 =i√2(Y 1
1 + Y 1−1) =
√
34π
y/r
pz = y10 = Y 02 =
√
34π
z/r
px = y11 = 1√2(Y 1
1 − Y 1−1) =
√
34π
x/r
dxy = y2−2 =i√2(Y 2
2 − Y 2−2) =
√
154π
xy/r2
dyz = y2−1 =i√2(Y 2
1 + Y 2−1) =
√
154π
yz/r2
d3z2−r2 = y20 = Y 02 =
√
154π
12√3(3z2 − r2)/r2
dxz = y21 = 1√2(Y 2
1 − Y 2−1) =
√
154π
xz/r2
dx2−y2 = y22 = 1√2(Y 2
2 + Y 2−2) =
√
154π
12
(x2 − y2)/r2
6.34 Eva Pavarini
B.3 Slater-Koster integrals
The interatomic Slater-Koster two-center integrals are defined as
Elm,l′m′ =
∫
drψlm(r− d)V (r− d)ψl′m′(r).
They can be expressed as a function of radial integrals Vll′α, which scale with the distance d
roughly as d−(l+l′+1) [38], and direction cosines, defined as
l = d · x/d, m = d · y/d, n = d · z/d.
The Slater-Koster integrals for s, p, and d orbitals [38] are listed below.